The Carry Trade: Risks and Drawdowns

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1 August 27, 2014 Comments Welcome The Carry Trade: Risks and Drawdowns Kent Daniel, Robert J. Hodrick, and Zhongjin Lu - Abstract - We examine carry trade returns formed from the G10 currencies. Performance attributes depend on the base currency. Dynamically spread-weighting and risk-rebalancing positions improves performance. Equity, bond, FX, volatility, and downside equity risks cannot explain profitability. Dollar-neutral carry trades exhibit insignificant abnormal returns, while the dollar exposure part of the carry trade earns significant abnormal returns with little skewness. Downside equity market betas of our carry trades are not significantly different from unconditional betas. Hedging with options reduces but does not eliminate abnormal returns. Distributions of drawdowns and maximum losses from daily data indicate the importance of time-varying autocorrelation in determining the negative skewness of longer horizon returns. Daniel: Graduate School of Business, Columbia University, and the NBER, Hodrick: Graduate School of Business, Columbia University, and the NBER, Lu: Terry College of Business, University of Georgia, This research was supported by a grant from the Network for Study on Pensions, Aging, and Retirement to the Columbia Business School. We especially thank Pierre Collin-Dufresne for many substantive early discussions that were fundamental to the development of the paper. We also thank Elessar Chen for his research assistance.

2 1 Introduction This paper examines some empirical properties of the carry trade in international currency markets. The carry trade is defined to be an investment in a high interest rate currency that is funded by borrowing a low interest rate currency. The carry is the ex ante observable positive interest differential. The return to the carry trade is uncertain because the exchange rate between the two currencies may change. The carry trade is profitable when the high interest rate currency depreciates relative to the low interest rate currency by less than the interest differential. 1 By interest rate parity, the interest differential is linked to the forward premium or discount. Absence of covered interest arbitrage opportunities implies that high interest rate currencies trade at forward discounts relative to low interest rate currencies, and low interest rate currencies trade at forward premiums. Thus, the carry trade can also be implemented in forward foreign exchange markets by going long in currencies trading at forward discounts and by going short in currencies trading at forward premiums. Such forward market trades are profitable as long as the currency trading at the forward discount depreciates less than the forward discount. Carry trades are known to have high Sharpe ratios, as emphasized by Burnside, Eichenbaum, Kleschelski, and Rebelo (2011). They are also known to do poorly in highly volatile environments, as emphasized by Bhansali (2007), Clarida, Davis, and Pedersen (2009), and Menkhoff, Sarno, Schmeling, and Schrimpf (2012). Brunnermeirer, Nagel, and Pedersen (2009) document that returns to carry trades have negative skewness. There is a substantive debate about whether carry trades are exposed to risk factors. Burnside, et al. (2011) argue that carry trade returns are not exposed to standard risk factors, in sample. Many others, cited below, find exposures to a variety of risk factors. Burnside (2012) provides a review of the literature. We contribute to this debate by demonstrating that non-dollar carry trades are exposed to risks, but carry trades versus only the dollar are not, at least not unconditionally in our sample. We also find that downside equity market risk exposure, which has recently been offered as an explanation for the high average carry trade profits by Lettau, Maggiori, and Weber (2014) and Dobrynskaya (2014) or in an alernative version by Jurek (2014), does not explain our carry trade returns. Finally, we conclude our analysis with a study of the drawdowns to carry trades. 2 define a drawdown to be the loss that a trader experiences from the peak (or high-water 1 Koijen, Moskowitz, Pedersen, and Vrugt (2013) explore the properties of carry trades in other asset markets by defining carry as the expected return on an asset assuming that market conditions, including the asset s price, stay the same. 2 Melvin and Shand (2014) analyze carry trade drawdowns including the dates and durations of the largest drawdowns and the contributions of individual currencies to the portfolio drawdowns. We 2

3 mark) to the trough in the cumulative return to a trading strategy. We also examine the pure drawdowns, as in Sornette (2003), which are defined to be persistent decreases in an asset price over consecutive days. We document that carry-trade drawdowns are large and occur over substantial time intervals. We contrast these drawdowns with the characterizations of carry trade returns by The Economist (2007) as picking up nickels in front of a steam roller, and by Breedon (2001) who noted that traders view carry-trade returns as arising by going up the stairs and coming down the elevator. Both of these characterizations suggest that negative skewness in the trades is substantially due to unexpected sharp drops. While we do find negative skewness in daily carry trade returns, the analysis of drawdowns provides a more nuanced view of the losses in carry trades. 2 Background Ideas and Essential Theory Because the carry trade can be implemented in the forward market, it is intimately connected to the forward premium anomaly the empirical finding that the forward premium on the foreign currency is not an unbiased forecast of the rate of appreciation of the foreign currency. In fact, expected profits on the carry trade would be zero if the forward premium were an unbiased predictor of the rate of appreciation of the foreign currency. Thus, the finding of apparent non-zero profits on the carry trade can be related to the classic interpretations of the apparent rejection of the unbiasedness hypothesis. The profession has recognized that there are four ways to interpret the rejection of unbiasedness forward rates: 1. First, the forecastability of the difference between forward rates and future spot rates could result from an equilibrium risk premium. Hansen and Hodrick (1983) provide an early econometric analysis of the restrictions that arise from a model of a rational, risk averse, representative investor. Fama (1984) demonstrates that if one interprets the econometric analysis from this efficient markets point of view, the data imply that risk premiums are more variable than expected rates of depreciation. 2. The second interpretation of the data, first offered by Bilson (1981), is that the nature of the predictability of future spot rates implies a market inefficiency. In this view, the profitability of trading strategies appears to be too good to be consistent with rational risk premiums. Froot and Thaler (1990) support this view and argue that the data are consistent with ideas from behavioral finance. 3. The third interpretation of the findings involves relaxation of the assumption that investors have rational expectations. Lewis (1989) proposes that learning by investors could reconcile the econometric findings with equilibrium theory. 3

4 4. Finally, Krasker (1980) argues that the interpretation of the econometrics could be flawed because so-called peso problems might be present. 3 Surveys of the literature by Hodrick (1987) and Engel (1996) provide extensive reviews of the research on these issues as they relate to the forward premium anomaly. Each of these themes plays out in the recent literature on the carry trade. Bansal and Shaliastovich (2013) argue that an equilibrium long-run risks model is capable of explaining the predictability of returns in bond and currency markets. Lustig, Roussanov, and Verdelhan (2014) develop a theory of countercyclical currency risk premiums. Carr and Wu (2007) and Jurek and Xu (2013) develop formal theoretical models of diffusive and jump currency risk premiums. Several papers find empirical support for the hypothesis that returns to the carry trade are exposed to risk factors. For example, Lustig, Roussanov, and Verdelhan (2011) argue that common movements in the carry trade across portfolios of currencies indicate rational risk premiums. Rafferty (2012) relates carry trade returns to a skewness risk factor in currency markets. Dobrynskaya (2014) and Lettau, Maggiori, and Weber (2014) argue that large average returns to high interest rate currencies are explained by their high conditional exposures to the market return in the down state. Jurek (2014) demonstrates that the return to selling puts, which has severe downside risk, explains the carry trade. Christiansen, Ranaldo, and Soderlind (2011) note that carry trade returns are more positively related to equity returns and more negatively related to bond risks the more volatile is the foreign exchange market. Ranaldo and Soderlind (2010) argue that the funding currencies have safe haven attributes, which implies that they tend to appreciate during times of crisis. Menkhoff, et al. (2012) argue that carry trades are exposed to a global FX volatility risk. Beber, Breedon, and Buraschi (2010) note that the yen-dollar carry trade performs poorly when differences of opinion are high. Mancini, Ranaldo, and Wrampelmeyer (2013) find that systematic variation in liquidity in the foreign exchange market contributes to the returns to the carry trade. Bakshi and Panayotov (2013) include commodity returns as well as foreign exchange volatility and liquidity in their risk factors. Sarno, Schneider and Wagner (2012) estimate multi-currency affine models with four dimensional latent variables. They find that such variables can explain the predictability of currency returns, but there is a tradeoff between the ability of the models to price the term structure of interest rates and the currency returns. Bakshi, Carr, and Wu (2008) use option prices to infer the dynamics of risk premiums for the dollar, pound and yen pricing kernels. 3 While peso problems were originally interpreted as large events on which agents placed prior probability and that didn t occur in the sample, Evans (1996) reviews the literature that broadened the definition to be situations in which the ex post realizations of returns do not match the ex ante frequencies from investors subjective probability distributions. 4

5 In contrast to these studies that find substantive financial risks in currency markets, Burnside, et al. (2011) explore whether the carry trade has exposure to a variety of standard sources of risk. Finding none, they conclude that peso problems explain the average returns. By hedging the carry trade with the appropriate option transaction to mitigate downside risk, they determine that the peso state involves a very high value for the stochastic discount factor. Jurek (2014), on the other hand, examines the hedged carry trade and finds positive, statistically significant mean returns indicating that peso states are not driving the average returns. Farhi and Gabaix (2011) and Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2013) argue that the carry trade is exposed to rare crash states in which high interest rate currencies depreciate. Jordà and Taylor (2012) dismiss the profitability of the naive carry trade based only on interest differentials as poor given its performance in the financial crisis of 2008, but they advocate simple modifications of the positions based on long-run exchange rate fundamentals that enhance its profitability and protect it from downside moves indicating a market inefficiency. 2.1 Implementing the Carry Trade This section develops notation and provides background theory that is useful in interpreting the empirical analysis. Let the level of the exchange rate of dollars per unit of a foreign currency be S t, and let the forward exchange rate that is known today for exchange of currencies in one period be F t. foreign currency interest rate be i t. 4 Let the one-period dollar interest rate be i $ t, and let the one-period In this paper, we explore several versions of the carry trade. The one most often studied in the literature is equal weighted in that it goes long (short) one dollar in each currency for which the interest rate is higher (lower) than the dollar interest rate. If the carry trade is done by borrowing and lending in the money markets, the dollar payoff to the carry trade without transaction costs can be written as z t+1 = [(1 + i t ) S t+1 S t (1 + i $ t )]y t (1) where y t = { +1 if i t > i $ t 1 if i $ t > i t } Equation (1) scales the carry trade either by borrowing one dollar and investing in the 4 When it is necessary to distinguish between the dollar exchange rate versus various currencies or the various interest rates, we will superscript them with numbers. 5

6 foreign currency money market, or by borrowing the appropriate amount of foreign currency to invest one dollar in the dollar money market. When interest rate parity holds, if i t > i $ t, F t < S t, the foreign currency is at a discount in the forward market. Conversely, if i t < i $ t, F t > S t, and the foreign currency is at a premium in the forward market. Thus, the carrytrade can also be implemented by going long (short) in the foreign currency in the forward market when the foreign currency is at a discount (premium)in terms of the dollar. Let w t be the amount of foreign currency bought in the forward market. The dollar payoff to this strategy is z t+1 = w t (S t+1 F t ) (2) To scale the forward positions to be either long or short in the forward market an amount of dollars equal to one dollar in the spot market as in equation (1), let w t = { 1 F t (1 + i $ t ) 1 F t (1 + i $ t ) } if F t < S t (3) if F t > S t When covered interest rate parity holds, and in the absence of transaction costs, the forward market strategy for implementing the carry trade in equation (2) is exactly equivalent to the carry trade strategy in equation (1). Unbiasedness of forward rates and uncovered interest rate parity imply that carry trade profits should average to zero. Uncovered interest rate parity ignores the possibility that changes in the values of currencies are exposed to risk factors, in which case risk premiums can arise. To incorporate risk aversion, we need to examine pricing kernels. 2.2 Pricing Kernels One of the fundamentals of no-arbitrage pricing is that there is a dollar pricing kernel or stochastic discount factor, M t+1, that prices all dollar returns, R t+1, from the investment of one dollar: E t [M t+1 R t+1 ] = 1 (4) Because implementing the carry trade in the forward market does not require an investment at time t, the no-arbitrage condition is E t (M t+1 z t+1 ) = 0 (5) 6

7 Taking the unconditional expectation of equation (5) and rearranging gives E(z t+1 ) = Cov(M t+1, z t+1 ) E(M t+1 ) (6) The expected value of the unconditional return on the carry trade could be non-zero if the dollar payoff on the carry trade covaries negatively with the innovation to the dollar pricing kernel The Hedged Carry Trade Burnside, et al. (2011), Caballero and Doyle (2012), Farhi, et al. (2013), and Jurek (2014) examine hedging the downside risks of the carry trade by purchasing insurance in the foreign currency option markets. To examine this analysis, let C t and P t be the dollar prices of one-period foreign currency call and put options with strike price K on one unit of foreign currency. Buying one unit of foreign currency in the forward market costs F t dollars in one period, which is an unconditional future cost. One can also unconditionally buy the foreign currency forward by buying a call option with strike price K and selling a put option with the same strike price in which case the future cost is K + C t (1 + i $ t ) P t (1 + i $ t ). To prevent arbitrage, these two unconditional future costs of purchasing the foreign currency in one period must be the same. Hence, F t = K + C t (1 + i $ t ) P t (1 + i $ t ) (7) which is put-call parity for foreign currency options. Now, suppose a dollar-based speculator wants to be long w t = (1 + i $ t )/F t units of foreign currency in the forward market, which, as above, is the foreign currency equivalent of one dollar spot. The payoff is negative if the realized future spot exchange rate of dollars per foreign currency is less than the forward rate. To place a floor on losses from a depreciation of the foreign currency, the speculator can hedge by purchasing out-of-the-money put options on the foreign currency. If the speculator borrows the funds to buy put options on w t units of foreign currency, the option payoff is [max(0, K S t+1 ) P t (1 + i $ t )]w t. The dollar payoff from the hedged long position in the forward market is therefore the sum of the forward purchase 5 Examples of such models include Nielsen and Saá-Requejo (1993) and Frachot (1996), who develop the first no arbitrage pricing models; Backus, Gavazzoni, Telmer and Zin (2010), who offer an explanation in terms of monetary policy conducted through Taylor Rules; Farhi and Gabaix (2011), who develop a crash risk model; Bansal and Shaliastovich (2013), who develop a long run risks explanation; and Lustig, Roussanov, and Verdelhan (2014), who calibrate a no-arbitrage model of countercyclical currency risks. 7

8 of foreign currency and the option payoff: z H t+1 = [S t+1 F t + max (0, K S t+1 ) P t (1 + i $ t )]w t Substituting from put-call parity gives z H t+1 = [S t+1 K + max (0, K S t+1 ) C t (1 + i $ t )]w t (8) When S t+1 < K, [S t+1 K+max (0, K S t+1 )] = 0; and if S t+1 > K, max(0, K S t+1 ) = 0. Hence, we can write equation (8) as z H t+1 = [max (0, S t+1 K) C t (1 + i $ t )]w t which is the return to borrowing enough dollars to buy call options on w t units of foreign currency. Thus, hedging a long forward position by buying out-of-the-money put options with borrowed dollars is equivalent to implementing the trade by directly borrowing dollars to buy the same foreign currency amount of in-the-money call options with the same strike price. Now, suppose the dollar-based speculator wants to sell w t units of the foreign currency in the forward market. An analogous argument can be used to demonstrate that hedging a short forward position by buying out-of-the-money call options is equivalent to implementing the trade by directly buying in-the-money foreign currency put options with the same strike price. When implementing the hedged carry trade, we examine two choices of strike prices corresponding to 10 and 25 options, where measures the sensitivity of the option price to movements in the underlying exchange rate. 6 Because the hedged carry trades are also zero net investment strategies, they must also satisfy equation (5). 3 Data In constructing our carry trade returns, we only use data on the G-10 currencies: the Australian dollar, AUD; the British pound, GBP; the Canadian dollar, CAD; the euro, EUR, spliced with historical data from the Deutsche mark; the Japanese yen, JPY; the New Zealand dollar, NZD; the Norwegian krone, NOK; the Swedish krona, SEK; the Swiss franc, CHF; and the U.S. dollar, USD. All spot and forward exchange rates are dollar denominated and are from Datastream and IHS Global Insight. For most currencies, the beginning of the sample is 6 The of an option is the derivative of the value of the option with respect to a change in the underlying spot rate. A 10 (25 ) call option increases in price by 0.10 (0.25) times the small increase in the spot rate. The of a put option is negative. 8

9 January 1976, and the end of the sample is August 2013, which provides a total of 451 observations on the carry trade. Data for the AUD and the NZD start in October Interest rate data are eurocurrency interest rates from Datastream. We explicitly exclude the European currencies other than the euro (and its precursor, the Deutsche mark), because we know that several of these currencies, such as the Italian lira, the Portuguese escudo, and the Spanish peseta, were relatively high interest rate currencies prior to the creation of the euro. At that time traders engaged in the convergence trade, which was a form of carry trade predicated on a bet that the euro would be created in which case the interest rates in the high interest rate countries would come down and those currencies would strengthen relative to the Deutsche mark. An obvious peso problem exists in these data because there was uncertainty about whether the euro would indeed be created. If the euro had not succeeded, the lira, escudo, and peseta would have suffered large devaluations relative to the Deutsche mark. We also avoid emerging market currencies because nominal interest rates denominated in these currencies also incorporate substantive sovereign risk premiums. The essence of the carry trade is that the investor bears pure foreign exchange risk, not sovereign risk. Furthermore, Longstaff, Pan, Pedersen, and Singleton (2011) demonstrate that sovereign risk premiums, as measured by credit default swaps, are not idiosyncratic because they covary with the U.S. stock and high-yield credit markets. Thus, including emerging market currencies could bias the analysis toward finding that the average returns to the broadly defined carry trade are due to exposure to risks. Our foreign currency options data are from JP Morgan. 7 After evaluating the quality of the data, we decided that high quality, actively traded, data were only available from September 2000 to August We also only had data for eight currencies versus the USD as option data for the SEK were not available. We describe the data on various risk factors as they are introduced below. Table A.1 in Appendix A provides distributional information on the risk factors. 4 Unconditional Results on the Carry Trade Table 1 reports basic unconditional sample statistics for five dollar-based carry trades from the G10 currencies. All of the statistics refer to annualized returns, and all the strategies invest in every currency in each period, if data are available. For the basic carry trade, labeled EQ, 7 We thank Tracy Johnson at JP Morgan for her assistance in obtaining the data. 9

10 the weight on currency j is w EQ j,t = sign(ij t i $ t ) N t where N t is the number of currencies. Thus, if the interest rate in currency j is higher (lower) than the dollar interest rate, the dollar-based investor goes long (short) 1/N t -th of a dollar in the forward market of currency j. This equal-weight approach is the way most academic articles implement the carry trade. Our second strategy is spread-weighted and is denoted SPD. The long or short position is again determined by the interest differential versus the dollar, but the fraction of a dollar invested in a particular currency is determined by the interest differential divided by the sum of the absolute values of the interest differentials: w SP D j,t = ij t i $ t N t i j t i $ t j=1 The SPD strategy thus invests more in currencies that have larger interest differentials while retaining the idea that the investment is scaled to have a total of one dollar spread across the absolute values of the long and short positions. Although it is often said that only one dollar is at risk in such a situation, this is not true when the trader shorts a foreign currency. When a trader borrows a dollar to invest in the foreign currency, the most the trader can lose is the one dollar if the foreign currency becomes worthless, but when a trader borrows the foreign currency equivalent of one dollar to invest in the U.S. money market, the amount of dollars that must be repaid could theoretically go to infinity if the foreign currency strengthens massively versus the dollar, as inspection of equation (1) indicates. Traders would consequently be unlikely to follow the EQ and SPD strategies because they take more risk when the volatility of the foreign exchange market is high than when it is low. In actual markets, traders typically face value-at-risk limits in which the possible loss of a particular amount on their portfolio of positions is calibrated to be a certain probability. For example, a typical value-at-risk model constrains a trader to take positions such that the probability of losing, say more than $1 million on any given day, is no larger than 1%. Implementing such a strategy requires a conditional covariance matrix of the returns on the nine currencies versus the dollar. To calculate this conditional covariance matrix, we construct simple IGARCH models from daily data. Let H t denote the conditional covariance matrix of returns at time t with typical element, h ij t, which denotes the conditional covariance between the i th and j th currency 10

11 returns realized at time t + 1. Then, the IGARCH model is h ij t = α(r i tr j t ) + (1 α)h ij t 1 (9) where we treat the product of the returns as equivalent to the product of the innovations in the returns. We set α = 0.06, as suggested in RiskMetrics (1996). To obtain the monthly covariance matrixes we multiply the daily IGARCH estimates of H t by 22. We use these conditional covariance matrixes in two strategies. In the first strategy, we target a fixed monthly standard deviation of 5%/ 12, which corresponds to an approximate annualized standard deviation of 5%, and we adjust the dollar scale of the EQ and SPD carry trades accordingly. These strategies are labeled EQ-RR and SPD-RR to indicate risk rebalancing. In the second strategy, we use the conditional covariance matrixes in successive one-month mean-variance maximizations. Beginning with the analysis of Meese and Rogoff (1983), it is often argued that expected rates of currency appreciation are essentially unforecastable. Hence, we take the vector of interest differentials, here labeled µ t, to be the conditional means of the carry trade returns, and we take positions wt OP T = ω t Ht 1 µ t, where H t is the conditional covariance matrix, and ω t is a scaling factor. We label this strategy OPT. As with the risk-rebalanced strategies, we target a constant annualized standard deviation of 5%. Thus, the weights in our portfolio are w OP T t = ( 0.05/ 12 ) ( ) µ t Ht Ht 1 µ t (10) µ t If the models of the conditional moments are correct, the conditional Sharpe ratio would equal ( µ th 1 t µ t ) Table 1 reports the first four moments of the various carry trade strategies as well as their Sharpe ratios and first order autocorrelations. Standard errors are in parenthesis and are based on Hansen s (1982) Generalized Method of Moments, as explained in Appendix B. 9 For the full sample, the carry trades for the USD-based investor have statistically significant average annual returns ranging from 2.10% (0.47) for the OPT strategy, to 3.96% (0.91) for the EQ strategy, and to 6.60% (1.31) for the SPD strategy. The strategies also have impressive Sharpe ratios, which range from 0.78 (0.19) for the EQ strategy to 1.02 (0.19) for the SPD- RR strategy. As Brunnermeier, Nagel, and Pedersen (2009) note, each of these strategies is 8 Ackermann, Pohl, and Schmedders (2012) also use conditional mean variance modeling so their positions are also proportional to Ht 1 µ t, but they target a constant mean return of 5% per annum. Hence, their positions satisfy w AP S µ t. While their conditional Sharpe ratio is also ( µ th 1 ) 0.5, µ t their t = (0.05/12) H 1 µ t H 1 t µ t t scaling factor responds more aggressively to perceived changes in the conditional Sharpe ratio than ours. 9 Throughout the paper, when we discuss estimated parameters, standard errors will be in parentheses and t-statistics will be in square brackets. 11 t

12 significantly negatively skewed, with the OPT strategy having the most negative skewness of (0.34). Table 1 also reports positive excess kurtosis that is statistically significant for all strategies. The first order autocorrelations of the strategies are low, as would be expected in currency markets, and only for the EQ-RR strategy can we reject that the first order autocorrelation is zero. Of course, it is well known that this test has very low power against interesting alternatives. The minimum monthly returns for the strategies are all quite large, ranging from -4.01% for the OPT strategy to -7.26% for the SPD. The maximum monthly returns range from 3.21% for the OPT to 8.07% for the SPD. Finally, Table 1 indicates that the carry trade strategies are profitable on between 288 months for the EQ strategy and 303 months for the OPT strategy out of the total of 451 months. 10 Table 2 presents results for the various equal-weighted carry trades in which each currency is considered to be the base currency and the returns are denominated in that base currency. For each strategy, if the interest rate in currency j is higher (lower) than the interest rate of the base currency, the investor goes long (short) in the forward market of currency j as in the dollar-based EQ strategy. The average annualized profits of these carry trades range from 2.33% (1.02) for the SEK to 3.92% (1.50) for the NZD. These mean returns are all less than the 3.96% (0.88) for the USD although given their standard errors it is unlikely that we would be able to reject equivalence of the means. For all base currencies, the average profitability is statistically significance at the.06 marginal level of significance or smaller. The Sharpe ratios of the alternative base-currency carry trades are also smaller than the USD-based Sharpe ratio of 0.78 (0.19). The lowest Sharpe ratio of the alternative base-currency carry trades is 0.36 (0.19) for the JPY, and the highest is 0.71 (0.18) for the CAD. The point estimates of skewness for the alternative base-currency carry trades are all negative, except for the EUR, and the statistical significance of skewness is high for the JPY, NOK, SEK, NZD, and AUD. All of these alternative-currency-based carry trades have positive, statistically significant, excess kurtosis. Only the GBP-based carry trade shows any sign of first-order autocorrelation. The maximum gains and losses on these strategies generally exceed those of the USD-based strategy. Only the CAD and EUR have smaller maximum monthly losses that the USD-based strategy, and the maximum monthly losses for the JPY, SEK, NZD, and AUD carry trades exceed 10%. The alternative base-currency carry trades are also profitable on slightly fewer days than the USD-based EQ trade, and the percentage of profitable days for the NZD and AUD is slightly smaller than for the USD. 10 The strategies are all positively correlated. Correlations range from.63 for EQ and OPT to.90 for EQ and SPD. The correlations of EQ and EQ-RR and SPD and SPD-RR are.88 and.89, respectively. 12

13 5 Carry-Trade Exposures to Risk Factors This section examines whether the average returns to the carry trades described above can be explained by exposures to a variety of risk factors. We include equity market, foreign exchange market, bond market, and volatility risk factors. In each case we run a regression of a carry-trade return, R t, on returns on assets or portfolios, F t, that represent the sources of risks, as in R t = α + β F t + ε t (11) Because we use returns as risk factors, the constant term in the regression, α, measures the average performance of the carry trade that is not explained by unconditional exposure of the carry trades to the market traded risks included in the regression. 5.1 Equity Market Risks Table 3 presents the results of regressions of our carry-trade returns on the three Fama-French (1993) equity market risk factors: the excess market return, R m,t, proxied by the return on the value-weighted NYSE, AMEX, and NASDAQ markets over the T-bill return; the return on a portfolio of small market capitalization stocks minus the return on a portfolio of big stocks, R SMB,t ; and the return on a portfolio of high book-to-market stocks minus the return on a portfolio of low book-to-market stocks, R HML,t. 11 Although the Fama-French (1993) factors exhibit some modest explanatory power with nine of the 15 coefficients having t-statistics that are greater than 1.88, the overall impression is that these equity market factors essentially leave most of the average returns of the carry trades unexplained, as the exposures to the risk factors are quite small which allows that constants in the regressions to range from 1.83% with a t-statistic of 3.72 for the OPT strategy, to 5.55% with a t-statistic of 4.84 for the SPD-RR strategy. The largest R 2 is also only.05. These equity market risk factors clearly do not explain the average carry trade returns. 5.2 Pure FX Risks Table 4 presents the results of regressions of our carry-trade portfolio returns on the two pure foreign exchange market risk factors proposed by Lustig, Roussanov, and Verdelhan (2011) who sort 35 currencies into six portfolios based on their interest rates relative to the dollar interest rate, with portfolio one containing the lowest interest rate currencies and portfolio six containing the highest interest rate currencies. Their two risk factors are the average 11 The Fama-French risk factors were obtained from Kenneth French s web site which also describes the construction of these portfolios. 13

14 return on all six currency portfolios, denoted R F X,t, which has a correlation of 0.99 with the first principal component of the six returns, and R HML F X,t, which is the difference in the returns on portfolio 6 and portfolio 1 and has a correlation of 0.94 with the second principal component of the returns. The data are from Adrien Verdelhan s web site, and the sample period is 1983: :08 for 358 observations. Given its construction as the difference in returns on high and low interest rate portfolios, it is unsurprising that R HML F X,t has significant explanatory power for our carry-trade returns with robust t-statistics between 6.22 for the OPT portfolio to 8.85 for the EQ-RR portfolio. The R 2 s are also higher than with the equity risk factors, ranging between 0.13 and Nevertheless, this pure FX risk model does not explain the average returns to our strategies as the constant terms in the regressions remain highly significant and range from a low of 1.29% for the OPT portfolio to 3.60% for the SPD-RR portfolio. 5.3 Bond Market Risks Because exchange rates are the relative prices of currencies and relative rates of currency depreciation are in theory driven by all sources of aggregate risks in the stochastic discount factors of the two currencies, it is logical that bond market risk factors should have explanatory power for the carry trade as the bond markets must price the important risks of changes in the stochastic discount factor. Table 5 presents the results of regressions of the carry trade returns on the excess equity market return and the excess return on the 10-year bond over the one-month bill rate, which represents the risk arising from changes in the level of interest rates, and the difference in returns between the 10-year bond and the 2-year note, which represents the risk arising from changes in the slope of the term structure of interest rates. The bond market returns are from CRSP. The coefficients on both of the bond market factors are highly significant. Positive returns on the 10-year bond that are matched by the return on the 2-year note, which would be caused by unanticipated decreases in the level of the USD yield curve, are bad for the USD-based carry trades. Notice also that the coefficients on the two excess bond returns are close to being equal and opposite in sign indicating that positive increases the return on the two-year note are bad for the carry trades. Notice, though, that the R 2 s of the regressions remain between.02 and.05, as in the equity market regressions, and that the constants in the regressions indicate that the means of the carry trade strategies remain statistically significant after these risk adjustment, with values between 2.15% for the OPT strategy to 6.71% for the SPD strategy. 14

15 5.4 Volatility Risk To capture possible exposure of the carry trade to equity market volatility, we introduce the return on a variance swap as a risk factor. This return is calculated as R V S,t+1 = Ndays d=1 ( ln P ) 2 ( ) t+1,d 30 V IXt 2 P t+1,d 1 Ndays where Ndays represents the number of trading days in a month and P t+1,d is the value of the S&P 500 index on day d of month t + 1. Data for V IX are obtained from the web site of the CBOE. The availability of data on the V IX limits the sample to 283 observations. The regression results are presented in Tables 6 and 7. Because R V S,t+1 is an excess return, we can continue to examine the constant terms in the regressions to assess whether exposure of the carry trade to risks explains the average returns. When we add R V S,t+1 to the regressions in Tables 6 to 7, very little changes as the absolute values of the largest t-statistics associated with the coefficients on R V S,t+1 are 1.40 for the equity market risks specification and 1.76 for the bond market risks, and the constant terms in these regressions remain highly significant. The coefficients on R V S,t+1 are negative in the specifications with the equity and bond market risks indicating that the carry trades do tend to do badly when equity volatility increases, but this exposure is not enough to explain the profitability of the trades. 6 Dollar Neutral and Pure Dollar Carry Trades In the basic EQ carry trade analysis discussed above, we take equal sized positions in nine currencies versus the dollar, either long or short, depending on whether a currency s interest rate is greater than the dollar interest rate or less than the dollar interest rate. Because there are nine currencies, the EQ portfolio is always either long or short some fraction of a dollar versus some set of currencies. To develop a dollar-neutral carry trade, we exclude the dollar interest rate and calculate the median interest rate of the remaining nine currencies. We then take equal long (short) positions versus the dollar in the four currencies whose interest rates are greater (less) than the median interest rate, and we scale the size of the equal-weighted positions to be the same magnitude as that of EQ. We use EQ0 to denote this portfolio which has zero direct dollar exposure. The second columns of Panels A and B of Table 8 report the first four moments of the EQ0 carry trade as well as the Sharpe ratio and the first order autocorrelation. Panel A reports the full sample results, and Panel B reports the results over the later part of the sample when V IX data become available (1990: :08). Heteroskedasticity consistent standard errors 15

16 are in parenthesis. For ease of comparison, we list the same set of statistics for EQ in Column 1. The EQ0 portfolio has statistically significant average annual returns in both samples, 1.61% (0.58) for the full sample and 1.72% (0.72) for the later sample. While these average returns are lower than that of the EQ strategy, the volatility of the EQ0 strategy is also lower, and its Sharpe ratio is 0.49 (0.19) in the full sample and 0.52 (0.23) in the later sample. These point estimates are about 30% lower than the respective Sharpe ratios of the EQ strategy. The skewness of the EQ0 strategy is -0.47(0.19) and (0.28) for the full sample and the later sample, respectively. The autocorrelation of the EQ0 strategy is negligible, and the maximum losses are smaller than those of the EQ strategy. The next question is whether the EQ0 strategy is exposed to risks. Column 2 of Table 9 presents the results of regressions of the EQ0 returns on the three Fama-French (1993) equity market risk factors. Notice that EQ0 loads significantly on the market return and the HML factor, with t-statistics of 5.29 and 2.83, respectively. The loading on the market return explains approximately 30% of the average return, and the loading on the HML factor explains another 15% of the average return. The resulting constant in the regression has a t-statistic of 1.54, and the R 2 is.10. In comparison, the regression of EQ returns on the same equity risk factors has an estimated constant of 3.39 with a t-statistic of 3.76 and an R 2 of only.04. The Fama-French (1993) three factor model clearly does a better job of explaining the average return of EQ0 strategy. These results are surprising for two reasons. First, the EQ0 strategy follows the same carry strategy using the G10 currencies as in the commonly studied EQ portfolio. The only difference is the dollar exposure, which makes an important difference in abnormal returns and risk factor loadings. Second, the literature currently leans toward the belief that FX carry trades cannot be explained by equity risk factors, but we find that the average returns to a typical FX carry trade can be explained by commonly used equity risk factors, with the caveat that the portfolio has zero direct exposure to the dollar. 6.1 A Decomposition of the Carry Trade To understand the difference between EQ and EQ0, we subtract the EQ0 positions from the EQ positions to obtain another portfolio, which we label EQ-minus. The currency positions in the EQ0 portfolio are w j,t = { + 1 N t 1 N t if i j t > med { } } i k t if i j t < med { } i k t where med { i k t } indicates the median of the interest rates excluding the dollar. 16

17 Subtracting the currency positions in the EQ0 strategy from those in EQ strategy gives the positions in EQ-minus, which goes long (short) the dollar when the dollar interest rate is higher (lower) than the median interest rate but only against currencies with interest rates between the median and dollar interest rates. The exact positions of the portfolio are the following: If i $ t < med { i k t }, then y j t = 2 N t 0 if i j t > med { } i k t if i j t = med { } i k t if i $ t < i j t < med { } i k t 0 if i j t i $ t 1 N t If i $ t > median { } i k t, then y j t = 2 N t 0 if i j t > i $ t if i j t = med { } i k t if med { } i k t < i j t i $ t 0 if i j t med { } i k t 1 N t The EQ0 and EQ-minus portfolios decompose the EQ carry trade into two components: a dollar neutral component and a dollar component. Column 3 of Panels A and B in Table 8 presents the first four moments of the EQ-minus strategy. which has statistically significant average annual returns in both samples, 2.35 (0.66) for the full sample and 2.11 (0.92) for the later sample. The Sharpe ratio of the EQ-minus strategy is higher than that of the EQ0 strategy in the full sample, 0.61 versus 0.49; but in later half of the sample, the EQ-minus strategy has a slightly lower Sharpe ratio, 0.49 versus 0.52, than the EQ0 strategy. Given the standard errors, one could argue that the Sharpe ratios are the same. Skewness of EQ-minus is quite negative (0.44) and (0.45) for the full sample and the later sample, respectively, although we note that the estimates of skewness are actually statistically insignificant due to the large standard error in both samples. In terms of the Sharpe ratio and skewness, the EQ-minus strategy appears no better than the EQ0 strategy. Nevertheless, the EQ-minus strategy has a correlation of -.11 with the EQ0 strategy, and the following results illustrate that the EQ-minus strategy also differs significantly from the EQ0 strategy in its risk exposures. Column 3 of Table 9 presents the results of regressions of the EQ-minus returns on the three Fama-French (1993) equity market risk factors. Unlike EQ0, only the SMB factor shows any explanatory power for the EQ-minus returns. The constant in the regression is 2.36% 17

18 with a t-statistic of The R 2 is.04. The equity market risk factors clearly do not explain the average returns to the EQ-minus strategy. Columns 1 to 3 of Table 10 present regressions of the returns to the EQ strategy and its two components, EQ0 and EQ-minus, on the equity market excess return and two bond market risk factors. Similar to our previous findings, the market excess return and the bond risk factors have significant explanatory power for the returns of EQ0 and a relatively high R 2 of.12. By comparison, none of the risk factors has any significant explanatory power for the returns on the EQ-minus strategy, and the resulting R 2 is only.02. Finally, Columns 1 to 3 of Table 11 present regressions of the returns of EQ and its two components, EQ0 and EQ-minus, on the two FX risk factors. The two factor FX model completely explains the average returns of the EQ0 strategy while explaining only 25% of the average returns of EQ-minus. The constant term in the EQ-minus regression also is significant with a value of 1.49% and a t-statistic of 1.86 in the FX two factor model. In summary, these results suggest that for the G10 currencies, conditional dollar exposure contributes more to the carry trade puzzle than does the non-dollar component. 6.2 The Pure Dollar Factor As noted above, the EQ-minus strategy goes long (short) the dollar when the dollar interest rate is above (below) the G10 median interest rate. It has the nice property of complementing the EQ0 strategy to become the commonly studied equally weighted carry trade. However, the other leg of the EQ-minus portfolio goes short (long) the currencies with interest rates between the G10 median rate and the dollar interest rate. It takes positions in relatively few currencies and thus seems under-diversified, which could explain its large kurtosis. Since the results just presented indicate that the abnormal returns of EQ hinge on the conditional dollar exposure, which is distinct from carry, we now expand the other leg of EQ-minus to all foreign currencies. We thus create the following strategy EQ-USD: { + y j 1 N t = if med { } } i k t > i $ t 1 if med { } i k N t i $ t The EQ-USD strategy focuses on the conditional exposure of the dollar. It goes long (short) the dollar against all nine foreign currencies when the dollar interest rate is higher (lower) than the global median interest rate. The fourth columns of Panels A and B of Table 8 present the first four moments of the returns to the EQ-USD strategy for the full sample and the sample for which equity volatility is available. We find that EQ-USD has statistically significant average annual returns in both samples, 5.54% (1.37) for the full sample and 5.21% 18

19 (1.60) for the later sample. Although its volatility is also higher than the EQ strategy, its Sharpe ratio of 0.68 (0.18) in the full sample and 0.66 (0.21) in the later sample are larger although not significantly different from those of the EQ strategy. Skewness of the EQ-USD strategy is insignificant as we find estimates of -0.11(0.17) for the full sample and (0.22) for the later sample. Thus, the EQ-USD strategy does not suffer from the extreme negative skewness often mentioned as the hallmark of carry trade. The fourth columns of Tables 9 and 11 report regressions of the returns of EQ-USD on the Fama-French (1993) three factors, the bond factors, and the FX risk factors. The only significant loading is on R F X,t, which goes long all foreign currencies. The constants in these regressions range from 5.18% with a t-statistic of 3.40 in the FX market risks regression to 5.82% with a t-statistic of 4.01 for the bond market risks regression. When we use all of the risk factors simultaneously in Table 12 for the shorter sample period, the foreign exchange risk factors and the volatility factor have significant loadings, but the constant in the regression is 4.52% with a a t-statistic of In summary, a large fraction of the premium earned by the EQ carry strategy can be attributed to its conditional dollar exposure. What is more, the EQ-USD portfolio built on this conditional dollar exposure earns a large premium, exhibits small exposures to standard risk factors, which cannot explain its return,and has insignificant negative skewness, indicating that negative skewness is not an explanation for the abnormal excess return of this strategy. 6.3 The Downside Market Risk Explanation Our last investigation of the potential risks of the carry trade considers two recent studies that offer exposure to downside equity market risk as the explanation of carry trade profitability. The first is by Lettau, Maggiori, and Weber (2014), and the second is by Jurek (2014) The Lettau, Maggiori, and Weber (2014) Analysis Lettau, Maggiori, and Weber (2014) first note that although portfolios of high interest rate currencies have higher market betas than portfolios of low interest rate currencies, these market-beta differentials are insufficiently large to explain the magnitude of carry trade returns. 12 To develop a market-return risk based explanation of the average carry trade returns, 12 Dobrynskaya (2014) uses a slightly different specification than Lettau, Maggiori, and Weber (2014) but reaches similar conclusions. Unfortunately, her GMM system of equations includes redundant orthogonality conditions as she includes the returns on 10 portfolios sorted on the interest differential as well as the 10-1 portfolio return differential. While this should result in a collinear set of orthogonality conditions, the 10-1 differential portfolio does not appear to be exactly equal to the difference in the returns on portfolio 10 minus portfolio 1 as the average returns are slightly different. If the covariance matrix of the orthogonality conditions is nearly singular, the resulting standard errors are not reliable. The reported efficient GMM estimates are also likely to be unreliable. Because the approaches are so similar, we focus our discussion on the approach taken by Lettau, Maggiori, and Weber (2014). 19

20 Lettau, Maggiori, and Weber (2014) modify the downside market risk model of Ang, Chen, and Xing (2006) and argue that the exposure of the carry trade to the return on the market is larger, conditional on the market return being down. In their empirical analysis, Lettau, Maggiori, and Weber (2014) define the downside market return, which we denote Rm,t, as equal to the market return when the market return is one standard deviation below the average market return and zero otherwise. 13 Lettau, Maggiori, and Weber (2014) find that the down-market-beta differential between the high and low interest rate sorted portfolios combined with a high price of down market risk is sufficient to explain the average returns to the carry trade. To examine whether the Rm,t risk factor explains our G10 carry trade strategies, we investigate whether our six currency portfolios have significant betas with respect to Rm,t and whether these downside betas are significantly different from the standard betas We show that our portfolios as well as other currency portfolios more generally, have statistically insignificant beta differentials which invalidates the explanation of Lettau, Maggiori, Weber (2014) for these carry trades. In their econometric analysis, Lettau, Maggiori, and Weber (2014) run separate univariate OLS regressions of portfolio returns, R t, on R m,t and Rm,t to define the risk exposures, β and β. Then, they impose that the price of R m,t risk is the average return on the market, E (R m,t ), and they use a cross-sectional regression to estimate a separate price of risk for Rm,t, denoted λ, which is necessary because Rm,t is not a traded risk factor. The unconditional expected return on an asset is therefore predicted to be E (R t ) = βe (R m,t ) + (β β)λ To test this explanation of our carry trades we first run a bivariate regression of a carry trade return on a constant; an indicator dummy variable, I, that is one when R m,t is non-zero and zero otherwise; and the two risk factors, R m,t and R m,t. Thus, we estimate R t = α 1 + α 2 I + β 1 R m,t + β 2 R m,t + ε t (12) In equation (12), β 1 measures the asset s basic exposure to the market excess return given that there is additional adjustment for when the market is in the downstate; and β 2 measures the asset s additional exposure to the market excess return when the market is in the downstate, that is β 2 = (β β 1 ). We then test the significance of β 2 using heteroskedasticity and 13 Because the definition of Rm,t used by Lettau, Maggiori, and Weber (2014) seemed arbitrary to us, we also considered two alternative more intuitive definitions which are the market return being less than the unconditional average market return and the market return being negative. The results are similar to the reported results and are available in an internet appendix. 20

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